Well-posedness and critical index set of the Cauchy problem for the coupled KdV-KdV systems on $ \mathbb{T} $

Abstract

<p style='text-indent:20px;'>Studied in this paper is the well-posedness of the Cauchy problem for the coupled KdV-KdV systems</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1100000"> \begin{document}$ \begin{equation} \left\{\begin{array}{rcl} u_t+a_{1}u_{xxx} &amp; = &amp; c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \\ v_t+a_{2}v_{xxx}&amp; = &amp; c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x}, \\ \left. (u, v)\right |_{t = 0} &amp; = &amp; (u_{0}, v_{0}) \end{array}\right. \ \ \ \ \ \ \ \ \left( {0.1} \right) \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>posed on the periodic domain <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{T} $\end{document}</tex-math></inline-formula> in the following four spaces</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{split} { \mathcal H}^s_1: = H^s_0 (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad { \mathcal H}^s_2: = H^s_0 ( \mathbb {T})\times H^s(\mathbb{T}), \\ { \mathcal H}^s_3: = H^s (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad { \mathcal H}^s_4: = H^s (\mathbb{T})\times H^s (\mathbb{T}). \end{split} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>The coefficients are assumed to satisfy <inline-formula><tex-math id="M3">\begin{document}$ a_1 a_2\neq 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \sum\limits_{i, j}(c_{ij}^2+d_{ij}^2)&gt;0 $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>Fix <inline-formula><tex-math id="M5">\begin{document}$ k\in\{1, 2, 3, 4\} $\end{document}</tex-math></inline-formula>. Then for any coefficients <inline-formula><tex-math id="M6">\begin{document}$ a_1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ a_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ (c_{ij}) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ (d_{ij}) $\end{document}</tex-math></inline-formula>, it is shown that there exists a critical index <inline-formula><tex-math id="M10">\begin{document}$ s^*_k \in (-\infty, +\infty] $\end{document}</tex-math></inline-formula> such that system (0.1) is analytically locally well-posed in <inline-formula><tex-math id="M11">\begin{document}$ \mathcal{H}^s_k $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M12">\begin{document}$ s&gt;s^*_k $\end{document}</tex-math></inline-formula> but weakly analytically ill-posed if <inline-formula><tex-math id="M13">\begin{document}$ s&lt;s^{*}_k $\end{document}</tex-math></inline-formula>. Viewing <inline-formula><tex-math id="M14">\begin{document}$ s^*_k $\end{document}</tex-math></inline-formula> as a function of the coefficients, its range <inline-formula><tex-math id="M15">\begin{document}$ \mathcal {C}_k $\end{document}</tex-math></inline-formula> is defined to be the <i>critical index set</i> for the analytical well-posedness of (0.1) in <inline-formula><tex-math id="M16">\begin{document}$ \mathcal {H}^s_k $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>By investigating some properties of the <i>irrationality exponents</i> of the real numbers and by establishing some sharp bilinear estimates in non-divergence form, we manage to identify</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ { \mathcal C}_1 = \left \{ -\frac12, \infty \right\} \bigcup \left \{ \alpha: \frac12\leq \alpha\leq 1 \right \} , \quad { \mathcal C}_q = \left \{ -\frac12, -\frac14, \infty \right\} \bigcup \left \{ \alpha: \frac12\leq \alpha\leq 1 \right \} . $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>for <inline-formula><tex-math id="M17">\begin{document}$ q = 2, 3, 4 $\end{document}</tex-math></inline-formula>. In particular, these sets contain an open interval <inline-formula><tex-math id="M18">\begin{document}$ (\frac12, 1) $\end{document}</tex-math></inline-formula>. This is in sharp contrast to the <inline-formula><tex-math id="M19">\begin{document}$ \mathbb {R} $\end{document}</tex-math></inline-formula> case in which the critical index set <inline-formula><tex-math id="M20">\begin{document}$ { \mathcal C} $\end{document}</tex-math></inline-formula> for the analytical well-posedness of (0.1) in the space <inline-formula><tex-math id="M21">\begin{document}$ H^s ( \mathbb {R})\times H^s ( \mathbb {R}) $\end{document}</tex-math></inline-formula> consists of exactly four numbers: <inline-formula><tex-math id="M22">\begin{document}$ { \mathcal C} = \left \{ -\frac{13}{12}, -\frac34, 0, \frac34 \right \}. $\end{document}</tex-math></inline-formula></p>